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International Journal of Mechanical Sciences
journalhomepage:www.elsevier.com/locate/ijmecsci
A novel differential kinematics model to compare the kinematic performances of 5-axis CNC machines
Chu A. My
a,∗, Erik L.J. Bohez
baDepartment of Special Robotics and Mechatronics, Le Quy Don Technical University, 236 Hoang Quoc Viet, Hanoi, Vietnam
bSchool of Engineering and Technology, Asian Institute of Technology, Klongluang, Pathumthani, Bangkok 12120, Thailand
a r t i c le i n f o
Keywords:
5-axis machine comparison Kinematic modeling Machine manipulability Machine dexterity Non-linear kinematic error
a b s t r a ct
A5-axisCNCmachineissimilartotwocooperatingrobots,onerobotcarryingtheworkpieceandonerobot carryingthetool.The5-axisCNCmachinesaredesignedinalargevarietyofkinematicconfigurationsand structures.Comparingdifferent5-axiskinematicconfigurationsplaysanimportantroleinmachineselection andoptimalmachinedesign.Inthissense,thepresentstudyproposesanewmathematicalmodeltoanalyze andcomparethekinematicperformancesofthe5-axismachines.First,ageneralizedkinematicchainof5-axis machineistreatedasaunifiedkinematicchainoftwocollaborativerobotsinordertoformulateageneralized differentialkinematicsmodelofthemachines.Second,fourimportantpropertiesofthekinematicsmodelare provedinageneralizedcasesothatquantitativeparameterscharacterizingthekinematicperformancesofthe machinescanbeevaluatedeffectively.Last,sixtypicalgroupsof5-axisCNCconfigurationsarecomparedthrough theevaluatedparameters.Inaddition,ithasbeenshownthat,byusingthepropertiesofthekinematicsmodel, theforwardandinversekinematicequationsfortherotaryaxesofany5-axismachinecanbeformulatedinan effectiveandsimplifiedmannerthatcouldbeusefulfordevelopingthepostprocessorsforany5-axismachine.
1. Introduction
Recently,5-axisCNCmachininghasbeenoneofthemostmodern andeffectivematerialremovaltechnologiesusedinmanufacturingin- dustries.The5-axisCNCmachineshavebeenusedformachiningtypical complexpartssuchasmolds,turbineblades,automotiveandaerospace partswhosegeometriesaretypicallydefinedbycomplexsurfaces.
A5-axisCNCmachineissimilartotwocooperatingrobots[1],one robotcarryingtheworkpieceandonerobot carryingthetool.Fig.1 showsthestructureandkinematicchaindiagramsofa5-axisCNCma- chineMaho600e.
Atypical5-axismechanismconsistsofthreeprismaticjoints(trans- lationalaxes)X,YandZ,andtworevolutejoints(rotaryaxes)AB,ACor BC.ThethreetranslationalaxesX,YandZrepresentthethreeorthog- onalmovementsalongwiththreeaxesofamachinecoordinatesystem (aCartersiancoordinatesystemfixedtothemachinebase)whoseZaxis isalways coincideswiththetoolaxisof amachine. Therotaryaxes A,BandCcharacterizetherotationsofthemachinetableorthema- chinespindleheadaroundtheaxisX,YandZofthemachinecoordinate system,respectively.Notethatwhenarotaryaxiswhosecenterlineis parallelwithoneoftheaxesX,YandZiscalledtheorthogonalrotary axis.Incontrast,ifthecenterlineofarotaryaxisisinclinedatanangle, itiscalledthenon-orthogonalrotaryaxis.
∗Correspondingauthor.
E-mailaddress:[email protected](C.A.My).
Theoretically,bytakingintoaccounttheorderofjoints,therewillbe 5!possiblecombinationsofjointsequenceforeachtypeof5-axismech- anism.Furthermore,eachcombinationofjointsequencehas6possible configurationsthatconsistoftwocooperativekinematicchains.Con- sequently,thenumberofpossibleconfigurationsof5-axismechanism is 3×5!x6=2160.However,inpractice,therotaryaxesof amachine areusuallyimplementednearesteithertotheworkpieceortothetool.
Therefore,thenumberofthepossibleconfigurationsofeachmachine typeisreducedtosix:(i)twoconfigurationsconsistingofbothrevolute jointson theworkpiececarryingchain(e.g. XYZABandXYZBA),(ii) twoconfigurationsconsistingofbothrevolutejointsonthetoolcarry- ingchain,and(iii)twoconfigurationsconsistingofonerotaryaxison thetoolcarryingchainandonerotaryaxisontheworkpiececarrying chain.Asaconsequence,thenumberoffeasibleconfigurationsof5-axis mechanismisrecalculatedas3×6×6=108.
Itisclearthatthe5-axismachinescanbedesignedinalargevariety of kinematicconfigurations andstructures.Therefore,comparison of themachinesplaysanimportantroleinselectingsuitablemachinesfor applicationsinmanufacturingindustriesandindevelopingnew5-axis CNCmachines.
Inrecentyears,someeffortshavebeentakingplacetosynthesize, analyzeandcomparemulti-axisCNCmachines[3,1,4].YanandChen [3]presentedageneralmethodtogenerateallpossibleconfigurations
https://doi.org/10.1016/j.ijmecsci.2019.105117
Received20May2019;Receivedinrevisedform27August2019;Accepted27August2019 Availableonline29August2019
0020-7403/© 2019ElsevierLtd.Allrightsreserved.
Fig.1. The5-axisCNCmachineMaho600e[1,2].
of machiningcenterswhich haveup tosevenDOFs.Bohez[1]clas- sified,ingeneral, the5-axismachinesintofourmaingroupsandin- vestigatedtheiradvantagesanddisadvantages.However,theseinves- tigationshavenotdiscussedaboutthekinematicperformancesofthe machinesyet.Tutunea-FatanandBhuiya[4]comparedthe5-axisCNC machinesthroughthenonlinearityerrors.However,thisstudymainly focusedonthe5-axismachinetypewithtwoorthogonalrotaryaxesim- plementedonthespindleheadonly.Comparisonofthe5-axismachines withrespecttoimportantkinematiccharacteristicssuchasthemanipu- labilityofamachineandtheflexibilityofthetool-workpieceorientation hasbeenoverlooked.Notethatthemanipulabilityofamachineandthe flexibilityof thetool-workpieceorientationplayanimportantrolein comparingtheperformancesofthe5-axisCNCmachines.Themanipu- labilityofamachinecharacterizesthetendencyofchangesindexterity characteristicsalongwiththevarianceofthemotionoftheaxes,and itindicatesofhowclosethemachineconfigurationistothesingular- ity.Theflexibilityofthetool-workpieceorientationimplieshowand underwhichanglethetoolcanorientrelativetotheworkpieceinthe workspaceofamachine.Themorethemanipulabilityandtheflexibility ofthetool-workpieceorientationofamachineare,themorecomplex partsthemachinecanmachinewithahighperformance.A5-axisma- chineofhighdexterityiscapableofmachiningcomplexpartsconsisting numeroussculpturedsurfaceswithreducedsetuptimessothatitcanin- creasetheproductivityofamanufacturingsystem.
Tocomparethekinematicperformancesofthe5-axismachineseffec- tively,ageneralizeddifferentialkinematicsmodelofthemachinesisof- tenrequired.However,mostofthepreviousinvestigationsonthekine- maticmodelingof5-axismachinemainlyfocusedonindividualtypesof themachines.
Decadesago,therehavebeeneffortsworkingonthekinematicmod- elingofthe5-axismachineswithtwoorthogonalrotaryaxes.Xuetal.
[5]analyzedthe5-axiskinematicsmodelwiththepurposeofminimiz- ingtheangularaccelerationoftherotaryaxes.Munlinetal.[6]and MunlinandMakhanov[7]focusedonakinematicsmodelofthe5-axis machineMaho600ewheninvestigatingtheoptimizationofthecutter rotationsnearsingularpoints.LeeandShe[8]formulated kinematic equationsforthetable-tiltingmachines,thespindle-tiltingmachines andthetable/spindle-tiltingmachines.Xuetal.[9]developedakine-
maticsmodelincorporatedwiththetoolinclinationsforthemachine typeXYZAC.Faroukietal.[10]investigatedtheoptimaltoolorienta- tioncontrolwiththeuseoftheinversekinematicsfortherotaryaxesAC ontheworkpiececarryingchain,andABonthetoolcarryingchain.Lav- ernheetal.[11]concentratedonthekinematicbehavioroftheMikron millingcenter(XYZAC).Wuetal.[12]andYunetal.[13]investigated thekinematicmodelingofindividual5-axismachineswithbothorthog- onalrotaryaxes.Withthepurposeofpostprocessordevelopment,Jung etal.[14]and,BozandLazoglu[15]formulatedthekinematicequa- tionsforthetable-tiltingtype5-axismachinesaswell.
Inrecentyears,therehavealsobeensomestudiesthatfocusedon thekinematicmodelingofthe5-axismachineswhichconsistof non- orthogonalrotaryaxes.My[2]and,MyandBohez[16]investigateda kinematicsmodelofthenutatingtable5-axismachinesDMU50eand DMU70e forthepostprocessordevelopmentandthekinematicerror minimization.Sørby[17]and,SheandHuang[18]studiedtheforward andinversekinematicequationsforthenutatingtableandnutatingspin- dle5-axismachinesalso.Liuetal.[19]formulatedthekinematicequa- tionswiththepurposeofidentificationof geometricerrors ofrotary axesin5-axismachinetoolswithnon-orthogonalrotaryaxesontheta- ble.Wangetal.[20]investigatedthekinematicmodelingofa5-axis CNCmachinewithoneorthogonalrotaryaxisonthetableandonenon- orthogonalrotaryaxisonthetoolchain.
Apartfromtheaforementionedworks,therehavebeenattemptsthat emphasizedonthegeneralizationofthekinematicsmodelforthe5-axis CNCmachines[21–25].SheandLee[25]proposedapostprocessorfor general5-axismachines,usingthekinematicsmodule,whichaddedtwo rotarymovementsontheworkpiecetableandtworotarymovementson thespindle.Tutunea-FatanandFeng[24]derivedageneralcoordinate transformationmatrixforall5-axismachineswithtworotaryaxes.The modelwasthenusedtoverifythefeasibilityofthetworotaryjoints withinthekinematicschainofthreemaintypesof5-axisCNCmachines.
SheandChang[22]didfurtherresearchonthebasisof[23]byextend- ingtheinversekinematicssolutionfortranslationalmotionsinaunified form.YangandAltintas[23],andLiuetal.[21]presentedageneralized kinematicsmodelusingScrewtheory.Notethatthekinematicequations proposedin[25,22]wereexpressedintermsofsevengeneralizedco- ordinatessincetwomorerevolutejointswereaddedonthekinematic
chainofageneral5-axisCNCmechanism;thekinematicequationsin [21,23]wererepresentedintheformofaproductofexponentialfunc- tions.Withthepurposeof comparingthekinematicefficiencyofthe 5-axisCNCmachines,theuseofsuchkinematicsmodelstoformulate thedifferentialkinematicequationsandevaluatethekinematicperfor- mancesofageneral5-axisconfigurationischallenging.
Theaboveraisedcriticalissuesleadtothemotivationsofdeveloping anewmethodtoevaluateandcomparethekinematicperformancesof the5-axisCNCmachines.Inthispaper,ageneralizeddifferentialkine- maticsmodelofthe5-axisCNCmachinesisformulated,whereageneral mechanismof5-axisCNCmachinesistreatedasaclosedloopmecha- nismoftwocooperatingrobotarms.Fourimportantpropertiesofthe generalizedkinematicsmodelofthe5-axisCNCmachinesareproved inageneralizedcasesothatthemanipulabilityindex,thenon-singular rangeofthefivejointvariables,thedexterityindex,theconditionnum- ber,andthenon-linearkinematicerrorofthemachinescanbeevaluated effectively.Basedonsuchtheevaluationoftheindicators,sixtypical typesof5-axisCNCconfigurationsarecompared.Itwasdemonstrated that,withthefourimportantpropertiesproved,thekinematicsmodel formulatedinthisstudyisadvantageousandeffectivewhencompared withthepreviousmodels.Itwasalsoshownthatthecomparisonofthe machinesisusefulwhenselectingsuitablemachinesforgivenapplica- tions,especiallywhenanalysingnewconceptualdesignsofa5-axisCNC machine.Inaddition,byusingthepropertiesoftheproposedkinemat- icsmodel,theinversekinematicequationsforany5-axisCNCmachines arederivedinasimplifiedandgeneralizedmanner.Thisisveryuseful andeffectivewhenworkingonthekinematicperformanceanalysisas wellasthepostprocessordevelopmentforany5-axisCNCmachines.
2. Formulationofageneralizedkinematicsmodelofthe5-axis machines
Inthissection,thekinematicequationsatpositionlevelandveloc- itylevelareformulatedforageneralizedkinematicchainofthe5-axis CNCmachines.Inparticular,fourimportantpropertiesofthekinematic equationsareproved.Theseusefulpropertieswillbetakenfulladvan- tageswhenevaluating andcomparingthekinematicperformancesof the5-axisCNCmachines.
Letusconsidera general5-axismechanismthatconsistsof three prismaticjoints(X,YandZ)andtworevolutejoints(AB/AC/BC).This mechanismis atypeof5DOFsclosed-loopmechanismwhichissyn- thesizedwithtwocollaborativekinematicchains.Thetwochainsare constrainedviaaplannedtoolpathduringthemachiningprocess.A generalkinematicdiagramofthe5-axisCNCmachinesispresentedin Fig.2.
Let𝐪=[ 𝑞1 𝑞2 𝑞3 𝑞4 𝑞5 ]𝑇 (qi ∈{X,Y,Z,A,B,C}) de- noteavectorofthegeneralizedcoordinatesof thesystem, O0x0y0z0 isareferencecoordinatesystem,Otxtytztisatoolcoordinatesystem, andOwxwywzw isaworkpiececoordinatesystem.Thetoolcoordinate systemOtxtytztislocatedatthetooltipandorientedparallelwiththe machinecoordinatesystemO0x0y0z0.Theworkpiececoordinatesystem Owxwywzwisusuallyplacedontheworkpieceandorientedparallelwith O0x0y0z0aswell.
Intheviewpointofthemultibodysystemdynamics,themotionof eachjointofa5-axisCNCmachinecanbedescribedbyahomogeneous transformationmatrixasfollows:
𝐇𝑖( 𝑞𝑖)
=
||||
||||
| [𝐄 𝐭𝑖
0 1 ]
, foraprismaticjoint𝑖 [𝐒𝑖 𝛕𝑖
0 1
]
, forarevolutejoint𝑖
(1)
whereEisa3×3identitymatrix,andtiisatranslationvectordescribing themotionofaprismaticjointi.Whenqi=X,thevectorticanbewrit- tenas𝐭𝑖=[ 𝑋+𝑋0 0 0 ]𝑇.Whenqi=Yorqi=Zthevectortiis representedas𝐭𝑖=[ 0 𝑌 +𝑌0 0 ]𝑇or𝐭𝑖=[ 0 0 𝑍+𝑍0 ]𝑇, respectively.X0,Y0andZ0aretheinitialvaluesoftheprismaticjoint
Fig.2. Ageneralkinematicdiagramofthe5-axisCNCmachines.
variables(theinitialpositionofthepointOtin thecoordinatesystem O0x0y0z0).𝝉irepresentstheoffsetdistancebetweenthecenterlineofa revolutejointqi (A,B or C)andacorrespondingaxisofthemachine coordinatesystem(O0x0,O0y0orO0z0).TherotationmatrixSicharac- terizingthemotionofarevolutejointicanbewrittenasfollows:
𝐒𝑖( 𝑞𝑖=𝐴)
=
⎡⎢
⎢⎣
1 0 0
0 cos𝐴 −sin𝐴 0 sin𝐴 cos𝐴
⎤⎥
⎥⎦
(2)
𝐒𝑖( 𝑞𝑖=𝐵)
=
⎡⎢
⎢⎣
cos𝐵 0 sin𝐵
0 1 0
−sin𝐵 0 cos𝐵
⎤⎥
⎥⎦
(3)
𝐒𝑖( 𝑞𝑖=𝐶)
=
⎡⎢
⎢⎣
cos𝐶 −sin𝐶 0 sin𝐶 cos𝐶 0
0 0 1
⎤⎥
⎥⎦
(4)
Notethatifthecenterlineofarevolutejoint𝑞𝑖isinclinedatanangle 𝛼(thenon-orthogonalrotaryaxis𝑞𝑖),andamatrixSR(𝛼)representsthe rotationofthecenterline,thematrixSi(qi)mustbeadditionallymulti- pliedbySR(𝛼)andSR(−𝛼)intheleftandrightsidesofSi(qi),respec- tively.
InthereferenceframeO0x0y0z0,thecumulativetransformationma- tricesforthetoolcarryingchainandfortheworkpiececarryingchain arecalculatedasfollows,respectively:
𝐇0𝑡=𝐇𝑛+1
(𝑞𝑛+1
)𝐇𝑛+2
(𝑞𝑛+2
)...𝐇5
(𝑞5
) (5)
𝐇0𝑤=𝐇𝑛( 𝑞𝑛)
𝐇𝑛−1
(𝑞𝑛−1
)...𝐇1
(𝑞1
) (6)
Forthecasesinwhichthelastjointq5isarevolutejoint,thetrans- formationmatrixH5(q5)mustbemultipliedbyatransformationmatrix characterizingthedistancebetweenthetooltipandthecenterlineofthe jointq5.
Onemoreinterestingfeatureofthemechanismunderconsideration is thatifthetwokinematic chainsareunifiedthroughthereference frameO0x0y0z0,theclosedloopmechanismofthetwochainsbecomes aserialopenmechanismofasinglekinematicchain.Thejointsequence oftheunifiedmechanismisq1,q2,q3,q4andq5.Themotionofthetool relativetotheworkpieceisthusdescribedbythefollowingkinematic relationship:
𝐇𝑤𝑡=( 𝐇0𝑤)−1
𝐇0𝑡
=𝐇−11 ( 𝑞1
)....𝐇−1𝑛 ( 𝑞𝑛)
𝐇𝑛+1
(𝑞𝑛+1
)...𝐇5
(𝑞5
) (7)
Sincen∈{0,...,5},thematrixHwtcanberewrittenasfollows:
𝐇𝑤𝑡=𝚯1
(𝑞1
)𝚯2
(𝑞2
)𝚯3
(𝑞3
)𝚯4
(𝑞4
)𝚯5
(𝑞5
) (8)
where
𝚯𝑖( 𝑞𝑖)
=
||||
||||
| [𝐄 𝐓𝑖
0 1
]
, foraprismaticjoint𝑖 [𝐑𝑖 𝛕𝑖
0 1
]
, forarevolutejoint𝑖
(9)
Withrespecttoallthejointsofthetoolcarryingchain, 𝐓𝑖=𝐭𝑖, foraprismaticjoint𝑖
𝐑𝑖=𝐒𝑖, forarevolutejoint𝑖 (10) Withrespecttoallthejointsoftheworkpiececarryingchain, 𝐓𝑖=−𝐭𝑖, foraprismaticjoint𝑖
𝐑𝑖=𝐒𝑇𝑖, forarevolutejoint𝑖 (11) Eq.(11)impliesthatallthetransformationmatricesdescribingthe motionofthejointsoftheworkpiececarryingchainmustbeinversed becausetheunifiedkinematicchainstartsattheworkpieceandendsat thetool.
Finally,thekinematicsmodelofthemachineisformulatedasfol- lows:
𝐇𝑤𝑡=
[𝐑𝑤𝑡 𝐩𝑇
0 1
]
(12)
where𝐩𝑇 =[ 𝑥 𝑦 𝑧 ]𝑇 isthepositionofthetooltipinthework- piececoordinatesystemOwxwywzw.Thethreedirectioncosinesofthe toolaxisvectori,jandkarethethreeentriesofthelastcolumnofthe matrixRwt.
Let’sdenote𝐗=[ 𝑥 𝑦 𝑧 𝑖 𝑗 𝑘 ]𝑇 asthetoolposture, theforwardkinematicequationofthesystemcanbewrittenasfollows:
𝐗=𝐟(𝐪) (13)
InEq.(13),Xisthesocalledthecutterlocationpoint(CLpoint) whichisusuallycalculatedbyCAD/CAMsystemswhenplanningatool pathforthe5-axisCNCmachining.ToproduceaG-codesfileforcon- trollinganindividualmachine,thefollowinginversekinematicequation mustbesolvedforq.
𝐪=𝐟−1(𝐗) (14)
NotethatEqs.(13and14)arethekinematicequationsatposition levelthathaveoftenusedforpostprocessordevelopmentforthe5-axis machines.Withthepurposeofevaluatingthekinematicperformancesof the5-axismachines,boththekinematicequationsatpositionleveland thekinematicequationsatvelocitylevelarerequired.Unfortunately, withthesixdependent equationsinEq.(13),itisimpossible tofor- mulatetheinversedifferentialkinematicequationsthatcanbeusedto evaluatethekinematicperformancesofthemachines.Eq.(13)iscom- posedoffiveindependentequationsandonedependentequation,since i2+j2+k2=1.Therefore,Eq.(13)needstobetransformedintoasetof allfiveindependentkinematicequations.
Letquandqv be thejointvariablesoftheprimary revolutejoint andthesecondaryrevolutejoint,whereu<v and(u,v)∈{1,2,3, 4,5}.Notethattheconditionu<vimpliesthattherevolutejointqu alwaysprecedestherevolutejointqvinanyjointordersoftheunified kinematicchain.Forallconfigurationsoftheunifiedkinematicchain, thejointquisalwaysclosertothewotkpiecethanthejointqv,anditis calledtheprimaryrevolutejoint.Thejointqvisthesecondaryrevolute joint.
Let 𝐪𝑇=[
𝑋 𝑌 𝑍 ]𝑇,
and 𝐪𝑅=[
𝑞𝑢 𝑞𝑣 ]𝑇
bethevectorsofthethreeprismaticjointvariablesandthetworevolute jointvariables,respectively.
Table1
Constantmatrix𝚽.
q u A B C
𝚽
[ 0 1 0 0 0 1
] [
1 0 0 0 0 1
] [
1 0 0 0 1 0 ]
Let 𝐩𝑅=[
𝜙 𝜑 ]𝑇
denotetheorientationofthetoolaxis.
Theparameters𝜑and𝜙arethetwoindependentdirectioncosines selected fromthreedependentdirection cosines(i,jandk) ofthetool axisvector.Theparameters𝜑and𝜙mustbe selectedsothatbothof themareexpressedintermsofboththevariablesquandqv.
𝐩𝑅=𝚽[
𝑖 𝑗 𝑘 ]𝑇, (15)
where𝚽isaconstantmatrixinTable1.
Forexample,whentheprimaryrevolutejointquistheA-axis,and thesecondaryrevolutejointqv istheB-axis,𝜑=k=cosqucosqv,and 𝜙=j=−sinqucosqv.
RewritingEq.(13)inaformoffiveindependentequationsyields
𝐩=𝐠(𝐪), (16)
where 𝐩=[
𝐩𝑇 𝐩𝑅 ]𝑇, (17)
and 𝐪=[
𝐪𝑇 𝐪𝑅 ]𝑇. (18)
Thus,thedifferentialkinematicequationforthe5-axismachinescan bewrittenasfollows:
̇𝐩=𝐉̇𝐪, (19)
whereJ5×5istheJacobianmatrix.
𝐉= [𝜕𝐩𝑇
𝜕𝐪𝑇 𝜕𝐩𝑇
𝜕𝐪𝑅
𝜕𝐩𝑅
𝜕𝐪𝑇 𝜕𝐩𝑅
𝜕𝐪𝑅
]
=
[𝐉𝑇 𝑇 𝐉𝑇 𝑅 𝐉𝑅𝑇 𝐉𝑅𝑅
] (20)
Eq.(19)isthedifferentialkinematicequationthatrelatesthejoint velocities,thetoolvelocityandtheJacobianmatrixwhichcharacterize thestructureofthemachines.
Itisworthnotingthatallthe5-axismechanismshavesomeimpor- tantcommonfeaturesasfollows.
ThefirstfeatureisthatthethreetranslationalaxesX,YandZare orthogonaleachother,whicharealignedwiththreeaxesofthemachine coordinatesystemO0x0y0z0.Particularly,inthisstudy,thedefinedco- ordinate systemsOtxtytztandOwxwywzw arealwaysparallelwiththe machinecoordinatesystemO0x0y0z0.
Thesecondoneisthatthetoolaxisvectoralwayspointsinthedi- rectionoftheaxisOtztofthetoolcoordinatesystemOtxtytzt.
ThethirdoneisthattherotaryaxesA,BandCimplytherotations ofthemachinetableorthemachinespindleheadaroundtheaxesO0x0, O0y0andO0z0ofthemachinecoordinatesystemO0x0y0z0,respectively.
Withrespecttothethreefeaturesabovementioned,someimportant propertiesofthekinematicsmodelofthegeneralmachinecanbeproved thatareveryusefulwhenworkingonthemodelingandanalysisofthe machinekinematicperformances.
Property1. Forallconfigurationsof5-axisCNCmachine,theforward kinematicequationsforthetworotaryaxescanbeformulateddirectly withthetworotationmatrices,regardlessofwheretheprimaryandthe secondary rotaryjointsareinthegeneralizedkinematicchainof the 5-axisCNCmachines.
Inotherwords,thetoolorientationvectorpRcanbecalculatedwith Eq.(21),andthedirectioncosinesi,jandkarecalculatedwithEq.(22). Boththecalculationsareindependentofthethreeprismaticjointsvari- ablesX,YandZ.
𝐩𝑅=𝚽𝐑𝑢𝐑𝑣𝚪 (21)
[ 𝑖 𝑗 𝑘 ]𝑇
=𝐑𝑢𝐑𝑣𝚪 (22)
whereRuandRvarethetworotationmatricesdescribingthemotionof theprimaryrevolutejoint(qu) andthesecondaryrevolutejoint(qv), accordingly.𝚪=[ 0 0 1 ]𝑇.
Theuseofthispropertywillreducethecomputationalcomplexity whenformulatingandanalyzingtheJacobianmatrixJ,themanipulabil- ityindex,thedexterityindex,andtheconditioningindexforthe5-axis CNCmachinesthatwillbepresentedinthenextsection.
Proof. Basedontherulesoftheblockmatrixmultiplication,thefollow- ingblockmatrixmultiplicationscanbeobtained:
[𝐄 𝐓𝑖−1
0 1
][ 𝐄 𝐓𝑖
0 1
]
= [𝐄 𝐓𝑇
0 1
]
(23)
[𝐑𝑖 𝛕𝑖
0 1
][ 𝐄 𝐓𝑖+1
0 1
]
=
[𝐑𝑖 𝐑𝑖𝐓𝑖+1+𝛕𝑖
0 1
]
(24)
[𝐄 𝐓𝑖−1
0 1
][ 𝐑𝑖 𝛕𝑖
0 1
]
=
[𝐑𝑖 𝐓𝑖−1+𝛕𝑖
0 1
]
(25)
ItcanbeseenfromEq.(23)thatmultiplyingtwotranslationaltrans- formationmatricesresultsinatransformationmatrixinthesameform of the multiplied matrices. Eqs.(24 and 25) show that multiplying atranslationaltransformationmatrixwitharotationaltransformation matrix,inadifferentorder,therotationblockmatrixRiintherotational transformationmatrixisnottransformed.Consequently,Eq.(12)canbe rewrittenasfollows:
𝐇𝑤𝑡=
[𝐑𝑢 𝐓𝑅𝑢
0 1
][
𝐑𝑣 𝐓𝑅𝑣
0 1
]
=
[𝐑𝑢𝐑𝑣 𝐑𝑢𝐓𝑅𝑣+𝐓𝑅𝑢
0 1
]
=
[𝐑𝑢𝐑𝑣 𝐩𝑇
0 1
] (26)
InEq.(26)TRuandTRvarethevectorsyieldedbythemultiplications ofarotationaltransformationmatrixwiththetranslationaltransforma- tionmatrices,respectively.
Sincethetoolaxisvectorpointsintheaxisztofthecoordinatesystem Otxtytzt,itsdirectioncosinesi,jandkarethethreeentriesofthelast rowoftherotationmatrixRuRv.Therefore,
[ 𝑖 𝑗 𝑘 ]𝑇 =𝐑𝑢𝐑𝑣𝚪 (27)
SubstitutingEq.(27)intoEq.(15)yieldsEq.(21)andcompletesthe proof.
It is also important tonote that, pR calculated with Eq. (21) is afunctionof only𝐪𝑅=[ 𝑞𝑢 𝑞𝑣 ]𝑇, anditis independentof 𝐪𝑇 = [ 𝑋 𝑌 𝑍 ]𝑇.AsaconsequenceofProperty1,
𝐉𝑅𝑇=𝜕𝐩𝑅
𝜕𝐪𝑇 =𝟎 (28)
Eq.(28)isanimportantconsequenceofProperty1thatisveryuseful whencalculatingtheJacobiandeterminantpresentedlateron.
Property2. ThedeterminantoftheJacobianmatrixJofthegeneralized 5-axiskinematicsmodelcanbedirectly calculatedbyusingonlythe kinematicssub-modelofthetworotaryaxes.Inotherwords,
𝐷𝑒𝑡(𝐉)=𝐷𝑒𝑡( 𝐉𝑅𝑅)
. (29)
Fig.3. Sevensequencesofthejointvariables.
Toevaluatethekinematicperformancesofa5-axismachine,theJa- cobiandeterminantisoftenneeded.However,formulationoftheJaco- biandeterminantinageneralizedcaseischallenging,sincethematrix Jhasadimensionof5×5,andthedeterminantisafunctionofallfive jointvariables.Hence,thispropertyisimportantwhichmakesitpossi- bletoformulatetheJacobiandeterminantforthemachines.Byusing Property2,Det(J5×5)canbeformulatedasthedeterminantofamatrix withadimensionof2×2only.
Proof. AsdiscussedintheProofofProperty1,itisclearlyseenthatall thetransformationmatrices𝚯i(qi)inEq.(8)areexpressedasparticular blockmatrices.Hence,multiplyingtwoorthreesuccessivetranslational transformationmatrices,indifferentorders,yieldsamatrixinthesame formofthemultipliedmatrices.However,multiplicationofatransla- tionaltransformationmatrixwitharotationaltransformationmatrixis notcommutative.Thus,theresultofthematrixchainmultiplicationin Eq.(8)dependsonwherethetworotationaltransformationmatricesare inthematrixchain.Theoretically,thereexistsevendifferentsequences ofthejointvariables(Fig.3)thatcorrespondtosevendifferentresults ofthematrixchainmultiplication.
Thefamilyofthe5-axismachineswithbothrotaryaxesonthetable isrepresentedbythesequenceofthejointvariablesS1.ThesequenceS3 representsthegroupof5-axismachineswiththeprimaryrotaryaxison thetable,andthesecondaryoneonthespindlehead.ThesequenceS5 denotesthe5-axismachineswithbothrotaryaxesonthetoolchain.The sequenceS2representsforallthecasesinwhichtherotationaltrans- formationmatrix𝚯v(qv)isinbetweentwotranslationaltransformation matrices,meanwhilethematrix𝚯u(qu)isthefirstmatrixofthematrix chain.Similarly,asforthesequenceS6,boththematrices𝚯u(qu)and 𝚯v(qv)areinbetweenacoupleoftranslationaltransformationmatrices;
thesequenceS7impliesthat𝚯u(qu)and𝚯v(qv)areadjacent,butboth thematricesareinthemiddleofthematrixchain.
Withthesevenchainsofthetransformationmatrices,thesevenre- sultsof thematrixchainmultiplicationcan bearchivedaccordingly, wherethetooltipposition𝐩𝑇=[ 𝑥 𝑦 𝑧 ]𝑇isexpressedasfollows:
S1.𝐩𝑇 =𝐑𝑢𝐑𝑣𝐪𝑇+𝐑𝑢𝛕𝑣+𝛕𝑢
S2.𝐩𝑇 =𝐑𝑢𝐑𝑣𝐪𝑇−𝐑𝑢𝐑𝑣( 𝐓2+𝐓3
)+𝐑𝑢( 𝐓2+𝐓3
)+𝐑𝑢𝛕𝑣+𝛕𝑢 S3.𝐩𝑇 =𝐑𝑢𝐪𝑇+𝐑𝑢𝛕𝑣+𝛕𝑢+𝐑𝑢𝐑𝑣𝛕𝑡
S4.𝐩𝑇 =𝐑𝑢𝐪𝑇+( 𝐄−𝐑𝑢)(
𝐓1+𝐓2
)+𝐑𝑢𝛕𝑣+𝛕𝑢+𝐑𝑢𝐑𝑣𝛕𝑡
S5.𝐩𝑇 =𝐪𝑇+𝐑𝑢𝛕𝑣+𝛕𝑢+𝐑𝑢𝐑𝑣𝛕𝑡 S6.𝐩𝑇 =𝐑𝑢𝐑𝑣𝐪𝑇−𝐑𝑢𝐑𝑣(
𝐓1+𝐓3
)+𝐑𝑢𝐓3+𝐑𝑢𝛕𝑣+𝛕𝑢+𝐓1
S7.𝐩𝑇 =𝐑𝑢𝐑𝑣𝐪𝑇+(
𝐄−𝐑𝑢𝐑𝑣)(
𝐓1+𝐓2
)+𝐑𝑢𝛕𝑣+𝛕𝑢
(30)
Itisclearthat,exceptfromtheJacobiandeterminantofthefirstterm ofalltheexpressions(S1-S7),theJacobiandeterminantofalltheother termsequaltozerobecausetheJacobianmatrixofaconstantvectoris azeromatrix,andthedeterminantofaJacobianmatrixcontainingone zero-columnequalstozero.
Consequently,fortheexpressionsS1,S2,S6andS7,theJacobianJTT canbecalculatedwithEq.(31).ForS3andS4,JTTiscalculatedwith Eq.(32),andforS5,JTTiscalculatedwithEq.(33).
𝐉𝑇 𝑇 =𝜕𝐩𝑇
𝜕𝐪𝑇 =𝜕(𝐑𝑢𝐑𝑣𝐪𝑇)
𝜕𝐪𝑇
=𝐑𝑢𝐑𝑣 (31)
𝐉𝑇 𝑇 =𝜕𝐩𝑇
𝜕𝐪𝑇 =𝜕(𝐑𝑢𝐪𝑇)
𝜕𝐪𝑇
=𝐑𝑢 (32)
𝐉𝑇 𝑇 =𝜕𝜕𝐩𝐪𝑇
𝑇 =𝜕(𝐪𝑇)
𝜕𝐪𝑇
=𝐄 (33)
NotethatDet(Ru)=Det(Rv)=Det(E)=1.Hence,forallthecases 𝐷𝑒𝑡(
𝐉𝑇 𝑇)
=1 (34)
Ontheotherhand,thedeterminantof theblockmatrixJcanbe calculatedasfollows:
𝐷𝑒𝑡(𝐉)=𝐷𝑒𝑡(
𝐉𝑇 𝑇𝐉𝑅𝑅−𝐉𝑇 𝑅𝐉𝑅𝑇)
(35) SubstitutingEqs.(28)and(34)intoEq.(35)yieldsEq.(29)andcom- pletestheproof.
Property 3. The Jacobian determinant Det(J), a function fJ(q)of a multi-variablesvectorq,canbetransformedandexpressedintermsof onlyonejointvariableqv.Inotherwords,
𝐷𝑒𝑡(𝐉)=𝑓𝐽( 𝑞𝑣)
. (36)
Property3isimportantwhensolvingDet(J(q))=0forq.
Proof. It is worth to note that 𝐑′𝑢=𝐑𝑢𝚲𝑢, and 𝐑′𝑣=𝐑𝑣𝚲𝑣, where 𝚲uand𝚲vareconstantmatriceswhichcanbelookedupinthefollowing Table2.
BasedonEq.(21),theJaocobianmatrix𝐉𝑅𝑅=[𝜕𝐩𝑅
𝜕𝐪𝑅]2×2canbefor- mulatedasablockmatrixmultiplicationasfollows:
𝐉𝑅𝑅=[ 𝚽𝐑𝑢]
2×3
[ 𝚲𝑢𝐑𝑣𝚪 𝐑𝑣𝚲𝑣𝚪 ]
3×2 (37)
Since𝚽𝚽T=E2×2, 𝐉𝑅𝑅=[
𝚽𝐑𝑢𝚽𝑇]
2×2
[ 𝚽𝚲𝑢𝐑𝑣𝚪 𝚽𝐑𝑣𝚲𝑣𝚪 ]
2×2
=𝐇𝑢𝐊𝑣 (38)
Notethat𝐇𝑢=𝚽𝐑𝑢𝚽𝑇=[cos𝑞𝑢 ∓sin𝑞𝑢
±sin𝑞𝑢 cos𝑞𝑢]isanorthogonalmatrix formulatedwithrespecttoquonly.Therefore,Det(Hu)=1.
Table2
Constantmatrices𝚲uand𝚲v.
q i A B C
𝚲i
⎡ ⎢
⎢ ⎣
0 0 0
0 0 −1
0 1 0
⎤ ⎥
⎥ ⎦
⎡ ⎢
⎢ ⎣
0 0 1 0 1 0
−1 0 1
⎤ ⎥
⎥ ⎦
⎡ ⎢
⎢ ⎣
0 −1 0
1 0 0
0 0 0
⎤ ⎥
⎥ ⎦
𝐊𝑣=[ 𝚽𝚲𝑢𝐑𝑣𝚪 𝚽𝐑𝑣𝚲𝑣𝚪 ]isamatrixformulatedwithrespect toRv(qv)only.Therefore,thedeterminantDet(Kv)isafunctionofonly onevariableqv,Det(Kv)=fJ(qv).
Finally,Eq.(29)canbetransformedasfollows:
𝐷𝑒𝑡(𝐉)=𝐷𝑒𝑡( 𝐇𝑢)
𝐷𝑒𝑡( 𝐊𝑣)
=𝑓𝐽(
𝑞𝑣) (39)
Eq.(39)completestheproof.
Property 4. Forall 5-axismechanisms,theconditionnumberof the Jacobianmatrix,𝜅(J),doesnotdependsontheprimaryrevolutejoint variablequ.Particularly,theconditionnumberforallthemachineswith bothrotaryaxesimplementedonthetoolcarryingchain(thespindle– tiltingmachines)dependsononlyonejointvariableqv.
Withthepurposeofcomparingthekinematicperformancesofdif- ferent5-axisCNCmachines,theconditionnumber𝜅(J)mustbetaken intoaccount.Generally,𝜅(J)dependsonthevariationofallfivejoint variablesofamachine.Hence,comparing𝜅(J)ofalldifferent5-axisma- chinesischallenging.Therefore,theuseofthisimportantpropertywill makespossibletheevaluationandcomparisonoftheconditionnumbers forthemachines.
Proof. DuetothefactthatifthematrixJcanbefactorizedintotwo matrices,wherethefirstmatrixisanorthogonalmatrix,andthesecond oneisamatrixindependentofqu,theeigenvaluesofthesecondmatrix will bethe singularvalues of J(𝜎1÷𝜎5),andthecondition number 𝜅(J)=𝜎max/𝜎ministhusindependentofqu.
RevisitingEq.(30),fortheexpressionsS1,S2,S3,S4,S6andS7,the blockmatrixJcanbederivedandfactorizedwithEq.(40).ForS5,J canbefactorizedwithEq.(41).
𝐉=
[𝐑𝑢 𝟎 𝟎 𝐇𝑢
][ 𝐑𝑣 𝐋𝑣
𝟎 𝐊𝑣 ]
(40)
𝐉= [𝐄 𝟎
𝟎 𝐑𝑢 ][
𝐄 (
𝐑𝑢𝐌𝑣−𝐄𝐌𝑣) 𝐍−1𝑣
𝟎 𝐄
][ 𝐄 𝐌𝑣 𝟎 𝐍𝑣 ]
(41)
TheblockLvinEq.(40)iscalculatedasfollows:
S1.𝐋𝑣=[
𝚲𝑢𝐑𝑣𝐪𝑇+𝚲𝑢𝛕𝑣 𝐑𝑣𝚲𝑣𝐪𝑇 ] S2.𝐋𝑣=
[ 𝚲𝑢𝐑𝑣𝐪𝑇−𝚲𝑢𝐑𝑣( 𝐓2+𝐓3
)+𝚲𝑢( 𝐓2+𝐓3
) +𝚲𝑢𝛕𝑣 𝐑𝑣𝚲𝑣𝐪𝑇−𝐑𝑣𝚲𝑣(
𝐓2+𝐓3
) ]
S3.𝐋𝑣=[
𝚲𝑢𝐪𝑇+𝚲𝑢𝛕𝑣+𝚲𝑢𝐑𝑣𝛕𝑡 𝐑𝑣𝚲𝑣𝛕𝑡 ] S4.𝐋𝑣=[
𝚲𝑢𝐪𝑇+𝚲𝑢( 𝐓1+𝐓2
)+𝚲𝑢𝛕𝑣+𝚲𝑢𝐑𝑣𝛕𝑡 𝐑𝑣𝚲𝑣𝛕𝑡 ] S6.𝐋𝑣=
[ 𝚲𝑢𝐑𝑣𝐪𝑇−𝚲𝑢𝐑𝑣( 𝐓1+𝐓3
)+𝚲𝑢𝐓3
+𝚲𝑢𝛕𝑣 𝐑𝑣𝚲𝑣𝐪𝑇−𝐑𝑣𝚲𝑣( 𝐓1+𝐓3
) ] S7.𝐋𝑣=
[ 𝚲𝑢𝐑𝑣𝐪𝑇−𝚲𝑢𝐑𝑣( 𝐓1+𝐓2
) +𝚲𝑢𝛕𝑣 𝐑𝑣𝚲𝑣𝐪𝑇−𝐑𝑣𝚲𝑣(
𝐓1+𝐓2
) ]
(42)
TheblocksMvandNvinEq.(41)arecalculatedasfollows:
𝐌𝑣=[
𝚲𝑢𝐑𝑣𝛕𝑡 𝐑𝑣𝚲𝑣𝛕𝑡 ]
(43)
𝐍𝑣=[
𝚲𝑢𝐑𝑣𝚪 𝐑𝑣𝚲𝑣𝚪 ]
(44) Itisclearlyseenthat,inthematrixmultiplicationEq.(40),thefirst matrixisanorthogonalmatrix,andthelastmatrixisindependentof qu.Since𝜅(J)iscalculatedwithonlytheeigenvaluesofthelastmatrix, itisindependentofquaswell.TheEqs.(41),(43)and(44)showthat thelastmatrixinthematrixmultiplicationEq.(41)isdependentonqv only,notdependentonqu,X,YandZ.Moreover,becausethefirstmatrix isanorthogonalmatrix,andthesecondoneisatriangularmatrixwith onesonthemaindiagonal,thesingularvaluesofJisonlyaffectedby thelastmatrix.Therefore𝜅(J)isdependentonqvonly.Thiscompletes theproof.
3. Comparisonofthekinematicperformancesofthe5-axisCNC machines
Inthissection,alltheprovedpropertiesofthekinematicsmodelare madefullusetoevaluateandcomparethekinematicperformancesof thesixmaintypesof5-axisCNCmachines.
Thefirstmachinetype(TypeI)includesallthemachineswhoseboth rotaryaxesareimplementedontheworkpiececarryingchain,andthe axesareorthogonal[1,5,8,11,22,23].
TypeIIconsistsofthemachineswithboththerotaryaxesonthe workpiececarryingchain.Onerotaryaxisisorthogonalandtheother oneisanon-orthogonalrotaryaxis[1,2,6,7,16–19].
TypeIIIisasetofthe5-axisCNCmachinesconsistingofbothrotary axesimplemented on thetoolcarryingchain,andboth theaxesare orthogonal[4,8,10,22–24].
TypeIVisafamilyofthemachineswiththetworotaryaxesonthe toolcarryingchain,butonlyonerotaryaxisisorthogonal[4,22,18].
TypeVcoversallmachinesconsistingofonerotaryaxisonthetool carryingchain andone rotaryaxisontheworkpiececarryingchain.
Boththeaxesareorthogonal[1,8,10,22,23].
Type VI includesthe machines withone rotary axison the tool carryingchain andone rotaryaxisontheworkpiececarryingchain.
However,therotaryaxisonthetoolcarryingchainisnon-orthogonal [18,20,22,23].
Fig.4. Thesixmachinetypes[8,18].
ThesixtypesofthemachinesareshowninFig.4.
Toevaluatethekinematicperformancesofamachine,themanipula- bilityindex,thedexterityindex,theconditionnumber,thenon-singular rangeofthejointvariablesandthenon-linearkinematicerrorareeval- uatedascommonindicatorstocomparethemachines.
3.1. Manipulabilityindex
Inordertoquantifythekinematicefficiencyofanindustrialrobot, thetendencyofchangesindexteritycharacteristicsalongwiththevari- anceoftherevolutejointvariablesisofimportanceandshouldbeanal- ysed.Inparticular,thekinematicmanipulabilityindex,𝜔=√
𝐷𝑒𝑡(𝐉𝐉𝑇) playsanessentialroleinthekinematicalperformanceanalysissinceit indicatesofhowclosethemachineconfigurationistothesingularity.
Inthisstudy,themanipulabilityindexisevaluatedforallthetypesof 5-axisCNCmachines.
ByusingProperty2,themanipulabilityindex𝜔fora5-axisCNC machinecanbeformulatedasfollows:
𝜔=
√𝐷𝑒𝑡( 𝐉𝐉𝑇)
=𝐷𝑒𝑡(
𝐉𝑅𝑅) (45)
RecallingEq.(37),themanipulabilityindexisformulatedasfollows:
𝜔=𝐷𝑒𝑡([
𝚽𝐑𝑢]
2×3
[ 𝚲𝑢𝐑𝑣𝚪 𝐑𝑣𝚲𝑣𝚪 ]
3×2
)
(46)
Itisshownthat𝜔canbecalculateddirectlywithonlyrotationma- tricesRuandRvofagivenmachine,regardlessofotherjointvariables X,YandZ.Therefore,themanipulabilityindex𝜔ofdifferentmachines canbeevaluatedinaneffectiveandsimplifiedmanner.
Fig.5showsthemanipulabilityindexofthreedifferentmachines.
ThefirstmachineisamachineTypeIVwiththeorthogonalrotaryaxis Ct andnon-orthogonalrotaryaxisBtonthetoolchain(CtBt-nutating machine).ThesecondmachineisamachineTypeI(SpinnerU5-620) whichhasbothorthogonalrotaryaxesonthetable(BwCw-orthogo- nal).ThethirdoneisamachineTypeII(DMU50E)whosetheaxisBw isnon-orthogonal(BwCw-nutating).Theinclinationangleofthenon- orthogonalrotaryaxisis45°.
Inthismanner,forallthesixmachinetypes,themaximumvalue ofthemanipulabilityindex𝜔maxcanbecalculatedandcomparedeffec- tively.ThecomparisonispresentedinSection3.6.
3.2. Non-singularrangeofjointvariables
In order toprovide more insightful information on themanipu- lability of a machine, the non-singular range of the joint variables 𝐪=[ 𝑞1 𝑞2 𝑞3 𝑞4 𝑞5 ]𝑇 needstobetakenintoaccount.Ina non-singularregionofthejointvariables,amachineoperatesunderthe desirabledexteritycondition,withoutsingularities.Inotherwords,the largerthenon-singularityrangeofthejointvariablesis,themoreflexi- blethetoolofamachinecanbeoriented,andthekinematicefficiency ofthemachineisincreased.
Inthisstudy,wedefineΠasthenon-singularrangeofthejointvari- ablesofa5-axisCNCmachine.Actually,Πisthesolutionofthefollowing inequality.
𝐷𝑒𝑡(𝐉)>0 (47)
Inthegeneralcase,evaluationofΠischallengingsinceEq.(47)is amulti-variablesinequality.Fortunately,byapplyingProperty3, the inequalityisdependentononlyonejointvariableqv.Therefore,Πof agiven5-axisCNCmachinecanbeobtainedbysolvingthefollowing inequalityforqv.
𝑓𝐽( 𝑞𝑣)
>0 (48)
Forinstance,thenon-singularrangeofthejointvariablesΠofthe machineSpinnerU5-620(TypeI)are−𝜋
2 <Π<0and0<Π<𝜋2. Thus,forallthemachinetypes,theindicatorΠcanbeevaluatedand comparedeffectively.ThecomparisonisdetailedinSection3.6.
3.3. Dexterityindex
Thedexterityindexisameasureofa5-axismachinetoachievedif- ferent orientationsforeach pointwithin theworkspace. Similartoa robotmanipulator,theorientationofthetoolofa5-axismachinecan be describedbyarotationmatrixusingparameterssuchasEuleran- gles,Roll-Pitch-Yawangles,etc.Itcanbeobservedthat,whena5-axis machineoperates,theorientationofthetoolaxisisarchivedbythero- tationofthetworotaryaxes.Forthisreason,atiltangle𝛼andaroll angle𝛽shouldbeconsideredtocharacterizethetoolorientationrela- tivetotheworkpiecesincethetiltangle𝛼canbecalculatedeasilywith respecttothedisplacementofthesecondrotaryjointonly.Theangle𝛽 isdeterminedwiththedirectioncosinesofthetoolaxisintheworkpiece coordinatesystem.Fig.6showsthetiltandrollanglesinallthreecases:
Fig.5.Theindex𝜔ofthenutatingheadconfiguration(CtBt-nutating),themachineSpinnerU5-620(BwCw),andthemachineDMU50E(BwCw-nutating).
Fig.6. Thetiltangle𝛼androll𝛽angleofatoolaxis.
Table3
Theformulationsof𝛼and𝛽.
q u 𝛼 𝛽
A arccos ( 𝑖 ) arctan2( j, k ) B arccos ( 𝑗) arctan2( k, i ) C arccos ( 𝑘 ) arctan2( i, j )
qu=A,qu=Bandqu=C.Table3presentstheformulationof𝛼and𝛽for thethreegeneralcasescorrespondingly.
Itisclearlyseenthat,fora5-axismachine,thelargertherangeof𝛼 and𝛽is,themoreflexiblethetoolorientationrelativetotheworkpiece canbeobtained.Themoretheflexibilityofthetoolorientationofama- chineis,themorecomplexthepartscanbemachinedwiththemachine.
Thesmallerrangeof𝛼and𝛽decreasestheorientationcapabilityofa machine.
Theangles𝛼and𝛽canvarywithintherangeof(0÷2𝜋).Thusthe dexterityindexcanbedefinedasfollows:
𝐷=1 2
(Δ𝛼 2𝜋 +Δ𝛽
2𝜋 )
, (49)
whereΔ𝛼=𝛼max−𝛼minandΔ𝛽=𝛽max−𝛽minarethepossiblerangeof variationoftheangles𝛼and𝛽foreachpointoftheworkspace.
ThedexterityindexDcanvary withintherangeof (0÷1).Ifthe dexterityindexisequaltounitywewillsaythatthemanipulatorhas fulldexterityataparticularpointoranarea.Forexample,thedexterity indexDofthemachineSpinnerU620(TypeI)is0.75sincethepossible rangeofthetiltangleΔ𝛼=𝜋andtherangeΔ𝛽=2𝜋.However,forthe machineDMU50e(TypeII),thedexterityindexDis1sincethema- chinehasthesamepossiblerangeofrollangle,butalagerrangeoftilt angleΔ𝛼=2𝜋.
3.4. Conditionnumber
Theconditionnumber𝜅(J)∈[1,+∞)isanimportantindexwhichis oftenusedtodescribefirsttheaccuracy/dexterityofamanipulatorand, second,theclosenessofaposetoasingularity.Theconditionnumber approachestoinfinitywhenamachineoperatesnearasingularity.The conditionnumberisalocalpropertyforany5-axisCNCmachineasit dependsonJacobianmatrixJwhichisastructuralproperty.Sincethe conditionnumbercharacterizesanormoftheJacobianmatrix,itisa measureoftherelativeamplificationofthecomputedcutterposition- ingerror𝛿p,uponalineartransformation,Eq.(50),withrespectthe systematicpositioningerrors𝛿q.
𝛿𝐩=𝐉𝛿𝐪 (50)
𝛕=𝐉𝑇𝐅 (51)
AscanbeseenfromEq.(51),theJacobianmatrixrelatestheinput forces/torques𝝉andoutputforces/torquesFofa5-axismachine.Thus, thesocalledmechanicaladvantageofamachinecanbeinvestigated throughtheconditionnumberaswell.
Whenperforminganidenticalmachiningtask,whichmachineshav- ingalargervalueoftheconditionnumber𝜅(J)willhavealargerpo- sitioningerror𝛿patthetooltipandrequiremoreinputforces/torques 𝝉.Inthissense, thecondition numbershouldbeminimizedin order tomaintainasuitablepositioningaccuracyandthemechanicaladvan- tage.Whentheconditionnumberequalsanoptimalvalueofone,the manipulatoris describedasisotropic.Isotropicconfigurationshavea numberofadvantages,includinggoodservoaccuracy,noiserejection, andsingularityavoidance.Atanypointinthenon-singularrangeofthe jointvariables,thelowertheconditionnumberis,thebetteroperating conditionofa5-axismachineisreached.
Asforthe5-axisCNCmachines,evaluatingandcomparingthecondi- tionnumber𝜅(J)isachallengingtasksinceJisaJacobianmatrixoffive jointvariables.Nevertheless,bytakingfulladvantagesofProperty4, theconditionnumberforthe5-axismachinescanbe formulatedand evaluatedeffectively.
AccordingtoProperty4,theconditionnumbercan becalculated withthelargestandsmallesteigenvaluesofthelastmatrixofthematrix multiplicationsEqs.(41)and(40)forthemachinesTypeIIIandIV,and forothermachinetypes,respectively.
ForthemachinesTypeIIIandIV,thecomputedconditionnumbers exhibitsaperiodicbehaviorshowninFig.7,whichdoesnotdependon thedisplacementsofthemachineaxesX,Y,Zandqu.Itdependsonqv only.NotethatthemachineswithdifferentconstantdistanceLtfrom thetooltiptothepivotpointoftherotaryaxisqvwillhavedifferent conditionnumbercurvesasshowninFig.6also.Asthevalueof𝜅(J)is dependentononlyqv,foranypoint(X,Y,Z)inthedomainofthelinear jointvariables,theconditionnumberofallmachinesTypeIIIandIV doesnotchange.
Figs.8and9showthecurvesoftheconditionnumberofamachine TypeIandTypeIIrespectively.Theshapeofthecurvesissimilartothat ofthepreviousonesinFig.7.Oneimportantthingisthatthevalueof 𝜅(J)israpidlyincreasedwhenX,YandZareincreased(Figs.7and8).
Whenthemachineoperates nearthezeropoint(X=Y=Z=0)inthe jointspace,thevalueof𝜅(J)isminimized.
IncomparisonwiththemachinesTypeIIIandIV,theconditionnum- beroftheothermachinetypesismuchlarger.Moreover,thecondition numberof themachinesTypeIIIandIVdoesnot changewithin the wholedomainoflinearaxesX,YandZ.Forothermachines,whenthe
Fig.7. Theconditionnumber𝜅(J)foramachineTypeIII.
Fig.8.Theconditionnumber𝜅(J)foramachineTypeI(SpinnerU620).
Fig.9. Theconditionnumber𝜅(J)foramachineTypeII(DMU50e).
cutteroperates fatherandfartherfrom thezeropoint,thecondition numberislargerandlarger.
3.5. Non-linearkinematictoolpatherror
WhenCLdataaregeneratedbyaCAMsystemitisassumedthatthe toolpathbetweentwosuccessiveCLpoints isastraightlinerelative totheworkpiece.However,duetotherotaryaxesofthemachine,the
actualtoolpathbetween twoblocks intheNCprogramwillbe non- linearrelativetotheworkpiece,reducingtheaccuracyofthetoolpath.
Ifthedeviationbetweenthestraightlineandtheactualtoolpathis greaterthantheallowablelimitation,morecuttercontactpointsneed tobeinsertedinbetweenthetwopoints.Thedeviationduetothenon- linearkinematicbehaviorof amachineis calledthenon-linearkine- maticerror of thetoolpath. Toreducethe non-linear kinematicer- rorin 5-axisfreeformsurfacemachining,anumberof blocksG01is
